Introduction

Videogames are one of my favorite past times. As a player who participates in ranked play, I’ve always kept my eye on the forefront of global competitions. One of the most notable games to establish a global competitive scene backed by paid professionals was the real-time strategy game (RTS) Starcraft II (SC2). In 2013 the top 10 starcraft players by earnings made nearly four-million dollars from their combined winnings (“Winnings: 2019 - Liquipedia - the Starcraft Ii Encyclopedia” n.d.). Watching top-caliber players reflexes and control is astonishing to even seasoned videogame enthusiasts. At the 2019 StarCraft II World Championship Finale, many others and I packed into the arena to see what these professions could do firsthand.

‘(“Congrats to the Starcraft Ii Wcs Global Finals Champion! - Blizzcon” n.d.)’

‘(“Congrats to the Starcraft Ii Wcs Global Finals Champion! - Blizzcon” n.d.)’

The eye watering speeds they perform at is universally referenced in gaming terminology as actions per minute (APMS). Professionals take actions at such fastspeeds (high APMS) it becomes challenging to follow their overall strategy. So past pondering their sheer speed, I found it difficult to distinctly define what made these players professionals.

To learn more about what defines talent in SC2 this analysis we will explore in game metrics in attempts to explain rank in competitive mode. The dataset used was provided by ‘(“UCI Machine Learning Repository: SkillCraft1 Master Table Dataset Data Set” n.d.)’.

Goal

To model the response LeagueIndex a sample of player data from a 2013 ranked season of Starcraft will be explored. The predictors provided summarize in game performance metrics for a season by player (GameID). The modeling process will consider all the predictor variables and then trim down until only significant predictors remain. Variables will be vetted for multicolinearity and finally the model will be explored for to see if the BLUE assumptions hold.

The goal of this analysis will be to test the explanatory power of APMs and other predictors that are less commonly discussed.

Limitations of the Model

The multivariate regression model used for the midterm 2 portions of this study explores the linear estimation of mean response of LeagueIndex estimated by predictors in the design matrix \(X\).

The assumptions of this model’s explanatory power depends on the residual error being gaussian. Considering LeagueIndex is ordinal variable it is doubtful if not impossible for the residuals to be statistically normal.

A more suitable form of model for this regression would be based off a Polytonomous Logistic Regression for Ordinal Response (Proportional Odds Model) (“Ordinal Logistic Regression | R Data Analysis Examples” n.d.). These methods will be revisited for the final portion of this analysis.

The Data Exploring

This dataset is a sample of averaged in-game metrics of Starcraft II players who participate in 2013 ranked play. The variables are as follows:

##  [1] "GameID"               "LeagueIndex"          "Age"                 
##  [4] "HoursPerWeek"         "TotalHours"           "APM"                 
##  [7] "SelectByHotkeys"      "AssignToHotkeys"      "UniqueHotkeys"       
## [10] "MinimapAttacks"       "MinimapRightClicks"   "NumberOfPACs"        
## [13] "GapBetweenPACs"       "ActionLatency"        "ActionsInPAC"        
## [16] "TotalMapExplored"     "WorkersMade"          "UniqueUnitsMade"     
## [19] "ComplexUnitsMade"     "ComplexAbilitiesUsed"

The appendix covers each in depth but the following are highlighted as a preface.

LeagueIndex The levels of LeagueIndex range (1:8) corresponding to player ranks Bronze, Silver, Gold, Platinum, Diamond, Master, Grand Master. Visible to the player in game each medal bronze through master is subdivided into divisions 1-5, and once again divided with rank points ranging 0-100 where anything past Grandmaster LeagueIndex=7 is not subdivided and unbounded in terms of rank points (“Leagues: 2019 - Liquipedia - the Starcraft Ii Encyclopedia” n.d.). The rating system similar to a Elo rating system common in chess. Elo’s systems have a extreme value distribution also known as a Gumbel distribution [https://chance.amstat.org/2020/09/chess/]. Although that distribution would be problematic as a nonnormal response it would provide some much needed continuity by transforming players LeagueIndex into a more continuous experimental variable. Unfortunately there are either unavaible or would require far too much cleaning for the scope of this analysis.

The limitation of predicting this ordinal response will be revisited more specifically along the exploration, modeling, and the predictions.

The following are the icon’s earned for players who achieve related rank by the end of a given season. The legends for the following plots are styled to match.

‘(“Congrats to the Starcraft Ii Wcs Global Finals Champion! - Blizzcon” n.d.)’

Actions Per Minute (APMs) - APMs apply to variety of games but are common metric for analyzing proficiency of players at RTS games, its theorized skills like this provide a great advantage to players (???) . Action quickness alone does not capture the strategy or macro/micro skills so these additional predictors may add some unique color in hopes of further explaining what makes players skilled.

Perception Action Cycles (PACs) - are the circular flow of information between an organism and its environment where a sensor-guided sequence of behaviors are iteratively followed towards a goal (“Perception-Action Cycle - Models, Architectures, and Hardware | Vassilis Cutsuridis | Springer” n.d.). In this data-set PACs are aggregate of screen movements where PAC is a screen fixation of containing at least one action (“UCI Machine Learning Repository: SkillCraft1 Master Table Dataset Data Set” n.d.).

Cleanliness

The missing values are related exclusive to players with LeagueIndex equivalent to Professional Players (8). Where the 55 players with LeagueIndex==8 the age data is NA and the HoursPerWeek are 0. LeagueIndexes 1-7 are obtainable playing matches in the base game and ranking up by winning. To be a professional you would have to be part of a team that has no direct part in the broader match making system. This study is aiming to understand how players go from being average to good, less so elite to best. The 55 values associated with professionals will be dropped to resolve both issues.

Another issue with LeagueIndex is that LeagueIndex 1-6 may contain any number of players, while LeagueIndex 7-Grandmaster may only contain some set range of players targeted at 1000 total per region (“Leagues: 2019 - Liquipedia - the Starcraft Ii Encyclopedia” n.d.). Dropping LeagueIndex 7 would be a step towards normality. Considering this multivariate linear model is already hampered by its selected application on a ELO system, LeagueIndex==7 will be kept to preserve a potential insight into the larger population of Starcraft II players.

In addition to the missing values we have a clear error with the TotalHours of one player. \(GameID = 5140\) has 1,000,000 TotalHours that equates to 114 years of game time.

If we assumed one extra zeros was added the end of the player’s TotalHours it equates to 14 years of played time on a game that is only 10 years old as of a 2020. Removing two zeros equates to 1.4 years of played time, and 3 zeros in 51.1 days of played time both that seems just as realistic. There is not a clear path to extrapolate this player’s true TotalHours so their data will be dropped from the analysis. This was originally detected during modeling, but brought earlier into that analysis because how obviously unintentional this value is.

Finally, preforming basic a inspection on HoursPerWeek a max value of 168 was discovered. Considering there are 168 hours in a week its not plausible for a individual player to do this. There could be multiple players using this account making this possible. Another prospect is that this player is actually an AI like google’s DeepMind (“AlphaStar: Mastering the Real-Time Strategy Game Starcraft Ii | Deepmind” n.d.). Either way this observation will be kept because what is realistic cutoff for the hours per week is not apparent and after removing this observation the next max value is 140 which seems almost as unrealistic.

Its worth noting that dropping any amount of high hour outliers still far from combats all the the potential abnormalities encountered through the use of HoursPerWeek and TotalHours. Multiple players could be using any of the accounts even if either variable is not relatively large. Potentially exacerbating the left-extrema is that nothing prevents one player from smurfing multiple times. Smurfing is when a player makes a additional accounts (“What Is a Smurf Account? Everything You Need to Know | Lol-Smurfs” n.d.). A common reason for doing this is to dominate the competition until their Elo rating adapts to their actual skill level.

Converting Units

Some of the time averaged metrics are per SC2 timestamp while other are per milisecond. To make these metrics more interpretable each metric will be converted into seconds. There are roughly 88.5 timestamps per second so each metric in timestamps will be multiplied that as a coefficient (“UCI Machine Learning Repository: SkillCraft1 Master Table Dataset Data Set” n.d.). Some of the time averaged metrics are per milisecond. These will be transformed into seconds so the time units are completely uniform. Both of these transformations are linear and will not affect our model’s assumptions.

Summary Statistics and Plots

Gaussianity of the Response

When using Shapiro-Wilk W test on response LeagueIndex the null hypothesis that the sample comes from normally distribution can be rejected. Besides the obvious issues with performing a W test with an ordinal response with a potentially underlying Gumbel distribution, the response has a negative skew with a mean of 4.12. Further more there is no reason that LeagueIndexes are uniforming spaced in terms of overall rank or skill.

Correlation Plot

Visually we can see that LeagueIndex has a relatively strong correlation with APM, SelectByHotkeys, AssignToHotkeys, NumberofPACs, GapBetweenPACs, and Action Latency. Some of these predictors may be the best choices for model, although its worth noting at this point many of the predictor values also have fairly strong correlations within themselves which may cause multiplecolinearity in a model. This is not too surprising because many of these metrics capture rate of actions in slightly difference forms. For example APM and NumberOfPacs likely have a strict mathematical relationship where approximately. \[NumberOfPACs \approx APM*MatchDurationMinutes\] The slight differences between these metrics them could have some deep explanatory power but that level of explore ration is beyond the scope of this analysis.

The following columns will be dropped as they may confound with APMs, ActionLatency, GapBetweenPACs,1 NumberofPACS, SelectbyHotkey, and ActionsInPAC.

Focusing exclusively on APM fits into an Occam’s razor approach by minimizing \(span(X)\).

Visual Relations

Visually determining trends between the predictors and the response with a ordinal response is is best done with alternatives to scatter plots. Violinplots [^ViolinPlots] will be used to gauge the linearity in relation to the response and distribution with variable at the varying levels (“A Complete Guide to Violin Plots | Tutorial by Chartio” n.d.). The apendix covers more details cocerning why they were selected.

MinimapAttacks, HoursPerweek, TotalHours, MinimapRightClicks, ComplexUnitsMade, ComplexAbilitiesUsed all have very long right tails. In search of gaussian predictors the listed variables were considered transformations, but this would have affected the simplicity of the explanation.2

No Relationship

Age the mean age of 21.6531775 does not vary much across LeaguIndex such that there is no stark linear relationship. Although the variance at the highest level seems to be much narrower then that at the lower levels.

Bimodal

HourPerWeek has visibly little or no relation to LeagueIndex 1-4 where LeagueIndex 4-7 seems to have a visible linear trend resulting in 0.03 vs 0.25. If HoursPerWeek survives the model trimming, it’s bimodality may cause issues with the model’s assumption \(COV(Y)=\sigma^2I\).Workersmade,ComplexUnitsMades, and ComplexAbilityUsed both have similar differences between LeagueIndex 1-4 and 4-7 with the portions that have no relation and a linear relation swapped in comparison to HoursPerWeek.

Linear

TotalHours, MinimapRightClicks, TotalMapExplored, and UniiqueUnitsMade have a positive linear trend with the response. APM has a strong linear relationship with the response.

Root

AssignToHotkeys, UniqueHotkeys, and MinimapAttacks have a notable square root relationship with the response. This also may cause issues with the Gaussianity of the model’s residuals.

Model Specifications

Multivariant Regression Model Manual Model Iterations (\(\Omega\) to \(\omega\))

A model with all predictors will be made. Subsequently predictors will be dropped one by one until only predictors with significance of at least \(\alpha=5\%\) remain starting with \(lm_\Omega\) and ending with \(lm_\omega\). The results are as follows:

The initial model has many significant predictors. The \(lm_\Omega\) summary:

Summary of Starting Manual Stepwise Backward Model Selection \(lm_\Omega\)
term estimate std.error statistic p.value
(Intercept) 1.4609 0.1366 10.7 0.0000
Age -0.0023 0.0046 -0.5 0.6167
HoursPerWeek 0.0051 0.0016 3.1 0.0019
TotalHours 0.0002 0.0000 7.2 0.0000
APM 0.0123 0.0005 23.4 0.0000
AssignToHotkeys 13.4224 1.2323 10.9 0.0000
UniqueHotkeys 0.0005 0.0001 4.7 0.0000
MinimapAttacks 11.1976 1.3853 8.1 0.0000
MinimapRightClicks -0.7756 0.6249 -1.2 0.2147
TotalMapExplored 0.0059 0.0031 1.9 0.0598
WorkersMade 2.7113 0.4408 6.2 0.0000
UniqueUnitsMade 0.0000 0.0001 0.2 0.8100
ComplexUnitsMade 1.9539 2.5076 0.8 0.4359
ComplexAbilitiesUsed 1.4389 1.0069 1.4 0.1531

Age,UniqueUnitsMade,ComplexUnitsMade, MinimapRightClicks, and TotalMapExplored were removed in that order across the 5 iterations of the model. All remaining predictors were significant to the predetermined \(\alpha\). The final iteration provided:

After 5 iterations all the predictors were significant to the predetermined alpha level. The final iteration provided:

Summary of Final Manual Stepwise Backward Model Selection\(lm_\omega\)
term estimate std.error statistic p.value
(Intercept) 1.5073 0.0584 25.8 0.0000
HoursPerWeek 0.0053 0.0016 3.2 0.0012
TotalHours 0.0002 0.0000 7.3 0.0000
APM 0.0123 0.0005 24.4 0.0000
AssignToHotkeys 13.5401 1.2313 11.0 0.0000
UniqueHotkeys 0.0005 0.0001 5.2 0.0000
MinimapAttacks 11.1888 1.3551 8.3 0.0000
WorkersMade 2.7573 0.4331 6.4 0.0000
ComplexAbilitiesUsed 2.3409 0.7983 2.9 0.0034

Assessing Fit and Overall Significance

A test will be performed to see if the predictors provide statistically significant better model than \(Y=\overset{\_}{Y}+\epsilon\) where \(Y=LeagueIndex\) .Thus the null model \(H_o\) is there is no systematic structure to the response LeagueIndex given the predictors \(X\) such that \(\beta=0\). Our alternative \(H_a\) is there is some relation such that \(LeagueIndex=X\beta+\epsilon\), where X all other variables with the exception of index GameID. Without much surprise using 13 predictor variables results of a very small p-value of ~0. Over the iteration this does not change in a notable fashion across the other models as the last model also results in a p-value0. Thus all \(\Delta p=p_\Omega-p_\omega\) iterations of the model we can reject the null hypothesis suggesting that we should further investigate the explanatory power of our alternative hypothesis.

Our null model \(H_o\) is there is no systematic structure to the response LeagueIndex. Our alternative \(H_a\) is there is some relation such that \(LeagueIndex=X\beta+\epsilon\), where X all other variables with the exception of index GameID. Without much surprise using 13 predictor variables results of a very small p-value of ~0. Over the iteration this does not change in a notable fashion across the 4 other models as the last model also results in a p-value0. Thus all iterations of the model we can reject the null hypothesis suggesting that we should further investigate the explanatory power of our alternative hypothesis’s more simplistic appoarch.

Testing for Signifance Between Models

If both models had normal residuals a F-test could be used on \(lm_\Omega\) and \(lm_\omega\) to determine if the model’s have significantly different residuals. \(RSS_\Omega\) and \(RSS_\omega\) both have Shapiro-Wilk’s test statistics that reject the null at \(\alpha=5\%\) shown in a later section that examines the normality of each models’ residuals. Without gaussian residuals conducting a ANOVA will be a only practice exercise for the final where hypothesis are provided by:

\[ H_o:RSS_\omega = RSS_\Omega\] \[ H_a:RSS_\omega \neq RSS_\Omega\]

Performing an ANOVA test we find that there is not significant difference in the models at \(\alpha=5\%\) such that we cannot reject the \(H_o\).(ANOVA see table below).The implications not rejecting \(H_o\) is that regardless of trimming the predictor space by \(\Delta p\) predictors, \(lm_\omega\) is expected to produce comparable residuals with a 5% chance this is a result of the sampling. Additionally their \(adjR^2\) is barely different. Where \(adjR^2_\Omega=\) 0.462 and \(adjR^2_\omega=\) 0.4614. Further analysis will be conducted to see if the predictive and explanatory power of the models differ past the magnitude of their residuals.

Res.Df RSS Df Sum of Sq F Pr(>F)
3322 3748.29 NA NA NA NA
3327 3757.68 -5 -9.39 1.66 0.14

The confidence Intervals Table3 show only a few subtleties between models coefficients confidence intervals.

In the starting model Age, MinimapRightClicks, UniqueUnitsMade, ComplexUnitsMade, and ComplexAbilitiesUsed are all not significant based on their p-value and their confidence intervals straddle 0. Interesting enough ComplexAbilitiesUsed was initially above the alpha value for significance but made it to the final model. This could be because of the removal of a confunding variable. The magnitude of this shift is reflected in its delta value and delta_width.

For comparison two summary statistics were added as follows:

Where: \[delta=\frac{(UL_\omega+LL_\omega)-(UL_\Omega+LL_\Omega)}{(UL_\omega+LL_\omega)}=\frac{MeanCI_\omega-MeanCI_\Omega}{MeanCI_\omega}\] \[delta_{width}=\frac{(UL_\omega-LL_\omega)-(UL_\Omega-LL_\Omega)}{(UL_\omega-LL_\omega)}\]

Confidence Intervals Statistics at \(\alpha\) 0.05
Row.names LL_s UL_s LL_f UL_f mean_s mean_f delta delta_width
(Intercept) 1.193 1.729 1.393 1.622 1.461 1.507 0.03 -1.34
Age -0.011 0.007 NA NA -0.002 NA NA NA
APM 0.011 0.013 0.011 0.013 0.012 0.012 0.00 -0.04
AssignToHotkeys 11.006 15.838 11.126 15.954 13.422 13.540 0.01 0.00
ComplexAbilitiesUsed -0.535 3.413 0.776 3.906 1.439 2.341 0.38 -0.26
ComplexUnitsMade -2.963 6.870 NA NA 1.954 NA NA NA
HoursPerWeek 0.002 0.008 0.002 0.008 0.005 0.005 0.03 -0.01
MinimapAttacks 8.481 13.914 8.532 13.846 11.198 11.189 0.00 -0.02
MinimapRightClicks -2.001 0.450 NA NA -0.776 NA NA NA
TotalHours 0.000 0.000 0.000 0.000 0.000 0.000 0.00 0.00
TotalMapExplored 0.000 0.012 NA NA 0.006 NA NA NA
UniqueHotkeys 0.000 0.001 0.000 0.001 0.000 0.001 0.08 -0.02
UniqueUnitsMade 0.000 0.000 NA NA 0.000 NA NA NA
WorkersMade 1.847 3.575 1.908 3.607 2.711 2.757 0.02 -0.02

Hours Metrics

Upon closer examination of the remaining predictors its surprising that TotalHours and HoursPerWeek do not have a higher correlation. I did not expect both two make it to the final model.

Complex Units/Abilities

The notable change in ComplexAbilitiesUsed’s confidence interval mentioned between the \(lm_\Omega\) to \(lm_\omega\) is likely a result the removal of the variable ComplexUnitsMade. Upon reintroducing ComplexUnitsMade into the final model we find the following p-values for both predictors to be insignificant. This is a warning that we may not be able to distinguish the effects or vary them independently. This aligns with what is expected based on the mechanics of the game. A players must make complex units before they can use their complex abilities.

The way forward with the model is to continue leaving out ComplexUnitsMade because only making a unit in SC2 is far from a win condition. Complex units must be utilized within percise contexts to reap their full value. On the otherhand worker units reflected in the predictor WorkerUnitsMade, are units a player may produce and subsequently assign them to do indefinite valued added work. If not interrupted by an attacking force, worker units will continue to add value in the inform of the in-game economy without additional intervention from the player as long as the resource node is still abundant.

Using complex abilities, furthermore using them well, is generally much more important then making these complex units in masses. This also compliments the initial goal of the modeling to add flavor to APMs in a way that may reveal what actions are important and fortunately APM and this ComplexUnitsUsed these there are mostly orthogonal with cor 0.14.

ANOVA \(lm_\omega\) and \(lm_\omega+ComplexUnitsMade\)
term estimate std.error statistic p.value
ComplexAbilitiesUsed 1.560207 1.003410 1.554905 0.1200637
ComplexUnitsMade 3.108488 2.420861 1.284043 0.1992165
Res.Df RSS Df Sum of Sq F Pr(>F)
3327 3757.68 NA NA NA NA
3326 3755.82 1 1.86 1.65 0.2

AssigntToHotkeys and Actions Per Minute

Assign to hotkeys is a fairly highly correlated with APM that they warrant a further investigation. AssignToHotkeys was considered to be dropped along with the PAC related predictors prior but I imaged it added a significant flavor to what type of actions regardless of its potential for confounding.AssignToHotkeys in game is when a player assigns units or buildings to hotkeys. For example if the player has two armies they may select all of the units in army 1 and use the hotkey combination CTRL+1 to assign those units to hotkey 1 for future use. Then the same player can select their second army use Ctrl+2 to hotkey 2. This works for any command unit or building in game, allowing the player to on the fly reshape hotkeys based as assets are gained or lost.

Both variables will be removed from the model one at a time and then compared to the final model that contains both. In both subset models’ RSS are statistically significant different. In terms of the impact to \(adjR^2\), as APMs seem to a explain a significant amount more than AssignToHotkeys as the models have with \(adjR^2\) of 0.37 and 0.44 compared to the final model 0.46. To reaffirm the initial effort of dropping predictor for the final model persued in the next model.

ANOVA \(lm_\omega\) and \(lm_\omega-APM\)
Res.Df RSS Df Sum of Sq F Pr(>F)
3327 3757.678 NA NA NA NA
3328 4430.796 -1 -673.118 595.97 0
ANOVA \(lm_\omega\) and \(lm_\omega-AssignToHotkeys\)
Res.Df RSS Df Sum of Sq F Pr(>F)
3327 3757.678 NA NA NA NA
3328 3894.264 -1 -136.586 120.931 0

Predict

Evaluate the predictive power of the model – particularly, how effective does the model appear to be at making predictions of future observations or the mean response. How might these predictions be unreliable? What are the limits of the prediction power, and where do we fall into extrapolation? Does this point stick across two different models?

Structural Uncertainty in the Model

Without using cross validation we can see some basic issues with the fitted values of the each model reviewed. Initially we can start with one of more dominate predictors APM and see that based some players are excepted to have a LeagueIndex>>10 which simply does not exist. Another part of the models bias is that it does not place anyone below LeagueIndex= 2

Prelude to final

7 Describe your proposed research question for the final. How will you revise your original research question? What issues have you encountered so far? What assumptions do you think you need to (re-)evaluate?

For the final, the logistical regression will remodel the same problem with a different set of techniques and assumptions that fit the ordinal response.

The goal is for the analysis pull in additional regression techniques while still integrating the previous exploratory exercises.

Appendix

About Columns

Attribute Information:

  1. GameID: Unique ID number for each game (integer)
  2. LeagueIndex: Bronze, Silver, Gold, Platinum, Diamond, Master, GrandMaster, and Professional leagues coded 1-8 (Ordinal)
  3. Age: Age of each player (integer)
  4. HoursPerWeek: Reported hours spent playing per week (integer)
  5. TotalHours: Reported total hours spent playing (integer)
  6. APM: Action per minute (continuous)
  7. SelectByHotkeys: Number of unit or building selections made using hotkeys per timestamp (continuous)
  8. AssignToHotkeys: Number of units or buildings assigned to hotkeys per timestamp (continuous)
  9. UniqueHotkeys: Number of unique hotkeys used per timestamp (continuous)
  10. MinimapAttacks: Number of attack actions on minimap per timestamp (continuous)
  11. MinimapRightClicks: number of right-clicks on minimap per timestamp (continuous)
  12. NumberOfPACs: Number of PACs per timestamp (continuous)
  13. GapBetweenPACs: Mean duration in milliseconds between PACs (continuous)
  14. ActionLatency: Mean latency from the onset of a PACs to their first action in milliseconds (continuous)
  15. ActionsInPAC: Mean number of actions within each PAC (continuous)
  16. TotalMapExplored: The number of 24x24 game coordinate grids viewed by the player per timestamp (continuous)
  17. WorkersMade: Number of SCVs, drones, and probes trained per timestamp (continuous)
  18. UniqueUnitsMade: Unique unites made per timestamp (continuous)
  19. ComplexUnitsMade: Number of ghosts, infestors, and high templars trained per timestamp (continuous)
  20. ComplexAbilitiesUsed: Abilities requiring specific targeting instructions used per timestamp (continuous)

Why Violin Plots.

I decided to use Violin plots because I found with less tweaking they provided almost all the information I was looking for compared to scatter plots. Head to head a limitation of violin plots is that make it seems as though the LeagueIndex level size contains the same \(n\). The histogram earlier in this analysis shows clearly that \(n\) at each level of the LeagueIndex is not equal so choosing a tool on the basis of reiterating that point seems redundant. The benefit of Violin plots is that they provide a smoothed density plot at each LeagueIndex with a single point that represents the mean. This same thing could be done with scatter plots but I found it took much more staring and plot to plot variation.

The following is some head to head varieties plotting the data.

References

“A Complete Guide to Violin Plots | Tutorial by Chartio.” n.d. Accessed November 2, 2020. https://chartio.com/learn/charts/violin-plot-complete-guide/.

“AlphaStar: Mastering the Real-Time Strategy Game Starcraft Ii | Deepmind.” n.d. Accessed November 2, 2020. https://deepmind.com/blog/article/alphastar-mastering-real-time-strategy-game-starcraft-ii.

“Congrats to the Starcraft Ii Wcs Global Finals Champion! - Blizzcon.” n.d. Accessed October 26, 2020. https://blizzcon.com/en-us/news/23198508/congrats-to-the-starcraft-ii-wcs-global-finals-champion.

“Leagues: 2019 - Liquipedia - the Starcraft Ii Encyclopedia.” n.d. Accessed October 26, 2020. https://liquipedia.net/starcraft2/Battle.net_Leagues.

“Ordinal Logistic Regression | R Data Analysis Examples.” n.d. Accessed October 27, 2020. https://stats.idre.ucla.edu/r/dae/ordinal-logistic-regression/.

“Perception-Action Cycle - Models, Architectures, and Hardware | Vassilis Cutsuridis | Springer.” n.d. Accessed October 29, 2020. https://www.springer.com/gp/book/9781441914514.

“UCI Machine Learning Repository: SkillCraft1 Master Table Dataset Data Set.” n.d. Accessed October 26, 2020. https://archive.ics.uci.edu/ml/datasets/SkillCraft1+Master+Table+Dataset.

“What Is a Smurf Account? Everything You Need to Know | Lol-Smurfs.” n.d. Accessed November 2, 2020. https://www.lol-smurfs.com/blog/what-is-a-smurf-account.

“Winnings: 2019 - Liquipedia - the Starcraft Ii Encyclopedia.” n.d. Accessed October 26, 2020. https://liquipedia.net/starcraft2/Winnings/2019.

  1. An additional issues with this predictor is that it does not seem to line up with the time units in the description. Before and after the unit transformation GapBetweenPACs results in a mean is 11.3094444 hours.)↩

  2. A log transformation would be preferable, but enough observations by GameID that contain at least 0 in the related predictor entry a would have to drop observation containing \(-\infty\). If a transformation is pursued for the second-half of this analysis it will likely be a square root transformation↩

  3. Delta values are normalized by dividing by the mean of the model by related predictors confidence interval↩